Mathematics Advanced • Year 11 • Module 2 • Lesson 11
Graphs of Tangent and Cotangent
Build procedural fluency in identifying period, asymptotes, and intercepts of tangent and cotangent graphs (basic and transformed).
1. Quick recall
Three one-mark warm-ups. 1 mark each
Q1.1 Complete the period statements:
$y = \tan x$ has period ________ rad.
$y = \cot x$ has period ________ rad.
Q1.2 State the location of the vertical asymptotes (as $x = \ldots$, $n \in \mathbb{Z}$):
$y = \tan x$: $x = $ ________________
$y = \cot x$: $x = $ ________________
Q1.3 In one sentence, explain why $\tan x$ is undefined at $x = \pi/2$. (Hint: use the definition $\tan x = \sin x / \cos x$.)
2. Worked example — period and asymptotes of $y = \tan(2x)$
Every line of working has a reason on the right.
Problem. Find the period of $y = \tan(2x)$ and the equations of all vertical asymptotes.
Step 1 — Identify $b$.
In $y = \tan(bx)$, here $b = 2$.
Reason: only the coefficient of $x$ affects period; the leading $1$ does nothing.
Step 2 — Apply the tangent period formula.
Period = $\pi / |b|$ = $\pi / 2$.
Reason: tangent's natural period is $\pi$ (not $2\pi$) — Trap 01 in the lesson.
Step 3 — Find where the function is undefined.
Need $\cos(2x) = 0$, so $2x = \pi/2 + n\pi$.
$\therefore x = \pi/4 + n\pi/2$, $n \in \mathbb{Z}$.
Reason: the input of tan, not $x$ itself, must avoid odd multiples of $\pi/2$.
Step 4 — Sanity check.
Distance between consecutive asymptotes = $\pi/2$ = period. ✓
Conclusion. Period $\pi/2$; asymptotes $x = \pi/4 + n\pi/2$.
3. Faded example — fill in the missing steps
Find the period and asymptotes of $y = \cot(3x)$. Fill in each blank. 4 marks
Step 1 — Identify $b$.
$b = $ ________.
Step 2 — Apply the cotangent period formula.
Period = $\pi / |b|$ = ________.
Step 3 — Cotangent is undefined where $\sin(\ldots) = 0$.
Need $\sin(\;\;\;\;\;) = 0$, so ________ = $n\pi$, giving $x = $ ________.
Step 4 — Sanity check.
Distance between consecutive asymptotes = ________ = period. ✓
Conclusion. Period = ________; asymptotes at $x = $ ________.
4. Graduated practice
For each, state period and the asymptotes within one period (use exact form in terms of $\pi$). Show one line of reasoning.
Foundation — base graphs (4 questions)
| Q | Function | Period | Asymptotes (in [0, $2\pi$)) |
|---|---|---|---|
| 4.1 1 | $y = \tan x$ | ||
| 4.2 1 | $y = \cot x$ | ||
| 4.3 1 | $y = \tan x$, $x$-intercepts in $[0, 2\pi)$ | ||
| 4.4 1 | $y = \cot x$, $x$-intercepts in $[0, 2\pi)$ | ||
Standard — transformed graphs (6 questions)
Show at least one line of reasoning per question.
4.5 Find the period and asymptotes (general form) of $y = \tan(4x)$. 2 marks
4.6 Find the period and asymptotes (general form) of $y = \cot(x/2)$. 2 marks
4.7 State the period and asymptotes (in $[0, 2\pi)$) of $y = 3\tan x$. Does the coefficient $3$ change the period or the asymptotes? Justify. 2 marks
4.8 Find the period and asymptotes (general form) of $y = \tan(\pi x)$. 2 marks
4.9 Find the period and asymptotes (general form) of $y = \cot(2x)$. 2 marks
4.10 Sketch $y = \tan x$ for $-\pi < x < \pi$ on the axes below, marking each asymptote with a dashed vertical line and each $x$-intercept with a dot. 2 marks
Extension — combine concepts (2 questions)
4.11 Show algebraically that $\tan(x + \pi) = \tan x$ for all $x$ where $\tan x$ is defined. Use this to explain why the period of $y = \tan x$ is $\pi$ (not $2\pi$). 3 marks
4.12 The function $y = \tan(bx)$ has consecutive asymptotes at $x = \pi/8$ and $x = 3\pi/8$. Find $b$ (assume $b > 0$) and write the general asymptote formula. 3 marks
5. Self-check the easy 3
Tick the first three once you've verified your method.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1 — Periods
Both have period $\pi$. (Not $2\pi$ — Trap 01.)
Q1.2 — Asymptotes
$y = \tan x$: $x = \pi/2 + n\pi$ (where $\cos x = 0$). $y = \cot x$: $x = n\pi$ (where $\sin x = 0$).
Q1.3 — Why $\tan(\pi/2)$ is undefined
$\tan(\pi/2) = \sin(\pi/2) / \cos(\pi/2) = 1/0$, which is undefined (division by zero). Hence a vertical asymptote at $x = \pi/2$.
Q3 — Faded example: $y = \cot(3x)$
Step 1: $b = 3$. Step 2: Period $= \pi/3$. Step 3: $\sin(3x) = 0 \Rightarrow 3x = n\pi \Rightarrow x = n\pi/3$. Step 4: Asymptote spacing $\pi/3$ = period. ✓ Period = $\pi/3$; asymptotes $x = n\pi/3$.
Q4.1 — $y = \tan x$
Period $\pi$. In $[0, 2\pi)$: asymptotes at $x = \pi/2$ and $x = 3\pi/2$.
Q4.2 — $y = \cot x$
Period $\pi$. In $[0, 2\pi]$: asymptotes at $x = 0, \pi, 2\pi$ (these are the boundary lines; in strict $[0, 2\pi)$ only $0$ and $\pi$).
Q4.3 — $x$-intercepts of $\tan x$
$x = 0$ and $x = \pi$ (where $\sin x = 0$ but $\cos x \neq 0$).
Q4.4 — $x$-intercepts of $\cot x$
$x = \pi/2$ and $x = 3\pi/2$ (where $\cos x = 0$ but $\sin x \neq 0$).
Q4.5 — $y = \tan(4x)$
Period $= \pi/4$. Asymptotes: $4x = \pi/2 + n\pi \Rightarrow$ $x = \pi/8 + n\pi/4$.
Q4.6 — $y = \cot(x/2)$
$b = 1/2$, period $= \pi / (1/2) = 2\pi$. Asymptotes: $x/2 = n\pi \Rightarrow$ $x = 2n\pi$.
Q4.7 — $y = 3\tan x$
Period is still $\pi$; asymptotes at $x = \pi/2$ and $3\pi/2$ in $[0, 2\pi)$. The vertical scale factor $3$ stretches values vertically but cannot move asymptotes or change horizontal period — only the input $bx$ affects those.
Q4.8 — $y = \tan(\pi x)$
$b = \pi$, period $= \pi/\pi = 1$. Asymptotes: $\pi x = \pi/2 + n\pi \Rightarrow$ $x = 1/2 + n$. (Note: $x$ is dimensionless here, asymptotes are at half-integers.)
Q4.9 — $y = \cot(2x)$
Period $= \pi/2$. Asymptotes: $2x = n\pi \Rightarrow$ $x = n\pi/2$.
Q4.10 — Sketch $y = \tan x$ on $(-\pi, \pi)$
Asymptotes at $x = -\pi/2$ and $x = \pi/2$ (dashed). $x$-intercept at $(0, 0)$. Three branches in the open interval: in $(-\pi, -\pi/2)$ a branch rising from $-\infty$ near $-\pi/2^-$ down from the right (wait — increasing throughout) — increasing from $-\infty$ at $-\pi/2^-$, but actually the branch on $(-\pi, -\pi/2)$ rises from $0$ at $x = -\pi$ up to $+\infty$ at $x = -\pi/2^-$ — no: $\tan(-\pi) = 0$, then increases through $x$-axis and shoots up to $+\infty$ at $-\pi/2^-$. Middle branch on $(-\pi/2, \pi/2)$ rises from $-\infty$ through $(0, 0)$ to $+\infty$.
Q4.11 — Period of $\tan x$ is $\pi$
$\tan(x + \pi) = \dfrac{\sin(x + \pi)}{\cos(x + \pi)} = \dfrac{-\sin x}{-\cos x} = \dfrac{\sin x}{\cos x} = \tan x$. Because the function repeats every $\pi$, the period is $\pi$ (the minimum positive value with this property — half of sine/cosine's $2\pi$ because both numerator and denominator flip sign simultaneously).
Q4.12 — $y = \tan(bx)$ given asymptotes
Distance between consecutive asymptotes = period = $\pi/b$. Here $3\pi/8 - \pi/8 = 2\pi/8 = \pi/4$. So $\pi/b = \pi/4 \Rightarrow$ $b = 4$. General asymptote: $bx = \pi/2 + n\pi \Rightarrow$ $x = \pi/8 + n\pi/4$. (Matches the data.)